A Note on the Practicality of Maximal Planar Subgraph Algorithms

Abstract

Given a graph G, the NP-hard Maximum Planar Subgraph problem (MPS ) asks for a planar subgraph of G with the maximum number of edges. There are several heuristic, approximative, and exact algorithms to tackle the problem, but-to the best of our knowledgethey have never been compared competitively in practice. We report on an exploratory study on the relative merits of the diverse approaches, focusing on practical runtime, solution quality, and implementation complexity. Surprisingly, a seemingly only theoretically strong approximation forms the building block of the strongest choice.

Exact Algorithms for the Maximum Planar Subgraph Problem

“…Instead, a sufficiently large but in many practical cases small subset of constraints is identified by a (heuristic) separation procedure. Over the years, the performance of this approach was improved by strong preprocessing [7], finding multiple violated constraints in linear time [12], and good heuristics [11].…”

“…In addition, we generated a set of random regular [29] graphs that are expander graphs with high probability. In [11] it was observed that such graphs seem to be especially hard at least for the Kuratowski formulation. For formulations that allow multiple configurations, we determined the most promising one in a preliminary benchmark on a set of 1224 Rome and North graphs, as reported in the previous sections.…”

“…The only non-trivial algorithm in literature for exactly computing a maximum planar subgraph is based on integer linear programming and Kuratowski's characterization of planarity [26]. Since its inception over two decades ago, no other exact algorithm has been proposed, and only few related algorithmic advances improved its performance, see [11].…”

Cycles to the Rescue! Novel Constraints to Compute Maximum Planar Subgraphs Fast

“…Still, since it has these guarantees on integral solutions, it suffices to obtain an exact algorithm. Over the years, the performance of this approach was improved by strong preprocessing [4], finding multiple Kuratowski subdivision in linear time [11], and strong primal heuristics [10]. We use all these identically in all considered algorithms.…”

“…There are several practical heuristic approaches to tackle the problem [10]. However, MPS is MaxSNP-hard, i.e., there is an upper bound < 1 on the obtainable approximation ratio unless P = NP [2], and there are further limits known for specific algorithmic approaches [3,7].…”